4 REPORT—1858. 
truth of the first premiss; in other words, it must depend on the nature and value 
of the definitions. If, as had often been asserted, a mathematical definition was 
merely an arbitrary phrase employed to prevent useless repetition, it would be value- 
less as far as the deduction of truth was concerned. An arbitrary definition never 
could lead to absolute truth. But if the definitions were not merely an assumed 
form of words, if, as Mr. Hennessy believed, they were necessarily involved in that 
fundamental conception under which the mind contemplates the foundations of geo- 
metry,—the idea of space—then the conclusions would be, in every sense of the word, 
true. There were two ways in which the necessary truth of the definitions might be 
shown. The first, and by far the most important, was by a metaphysical analysis. 
This had been done successfully by Dr. Whewell. The second, which had not 
hitherto been attempted, was by a careful comparison of the definitions one with 
another. This Mr. Hennessy then proceeded to do. Taking the earliest examples 
with which we are acquainted, he showed that Euclid’s definition of a right line and 
his definition of a plane surface had the same differentia. Plato’s definitions bore 
a similar intimate relation to each other. Hero’s definition of a plane surface, and 
that adopted some centuries afterwards, of a right line, were also similar. He 
pointed out that such analogies may be traced throughout the whole group of defi- 
nitions, in which not only Euclid’s Geometry, but any mathematical work what- 
ever, as the mathematical Euclid, for example, is built up. Where the analogy ap- 
pears to fail, it will be found, on careful examination, that it only makes the case 
still stronger. For instance, Euclid’s definitions of a circle and of parallel lines ap- 
pear not to have the slightest analogy. This is readily explained by the fact that the 
latter definition, as every one admits, is defective. But when we substitute for the 
defective definition one which is sound and useful, an analogy becomes evident. 
Parallel lines may properly be defined to be two right lines, the perpendicular di- 
stances between which are all equal. A definition closely resembling this had been 
used by Boscovich and Wolfius. From it every property of parallels may be easily 
deduced. Euclid’s definition of a circle is, a continued line having a certain point 
within it from which all right lines drawn to the continued line are equal. Now all 
the radii of the circle are perpendicular to the circumference ; and if we could sup- 
pose one line, of two parallels, to become a point, the other would become a circle. 
The close analogy between the definitions of these two important conceptions is 
thus evident. He believed also that these definitions contained the germs of all the 
properties subsequently developed. If a definition were not a necessary truth, but 
were merely an arbitrary form of words employed to prevent repetition, such analo- 
gies as these never would have existed. Their existence shows that mathematical 
definitions are neither accidental nor mere matters of choice. 
On some Properties of a Series of the Powers of the same Number. 
By J. Pore Hennessy, of the Inner Temple. 
The author announced the discovery of some general laws which regulate the series 
of the powers of any number. For instance, in the following series of the powers 
of 5, the number of digits in the several recurrent vertical series may be expressed 
by the powers of 2 :— 
Number of digits recurring. 
5 See Ist. 
1 ew a = Sale 25 pe 2nd. 
2 Ee me ne aa 125 pe 3rd. 
625 oe 4th. 
4 a5 BS van <== 3125 due 5th. 
8 uae we ae Bs 15625 lige 6th. 
78125 wa 7th. 
16 Buia oh a3 =a 390625 Be 8th. 
32 ae 26 A ae 1953125 weé 9th. 
0765625 at 10th. 
64 ee iter aia see 48828125 i 11th. 
128 oan ee ae 5 54 244140625 dai 12th. 
256 wee eee vee ee 1220703125 = 13th. 
